Research
Gravitational radiation from a rotating magnetic dipole
aInstituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, Cd. de México, 01000, México.
Abstract
The gravitational radiation emitted by a rotating magnetic dipole is calculated. Formulas for the polarization amplitudes and the radiated power are obtained in closed forms. A comparison is made with other sources of gravitational and electromagnetic radiation, particularly neutron stars with extremely powerful magnetic fields.
Keywords: Gravitational waves; neutron stars
PACS: 04.30.Db; 04.30.Tv
1. Introduction
Gravitational radiation is an important source of energy in many astrophysical phenomena. A neutron star, for instance, radiates both electromagnetic 1,2 and gravitational waves 3,4,5; the main sources of radiation are the interior of the star (behaving as a magnetized fluid), the external magnetic field and the corotating magnetosphere.
In the present article, we study the gravitational waves (GWs) generated by one of these possible sources: a rotating magnetic dipole. In Sec. 2, the radiated energy and the polarization amplitudes of the GWs are calculated using the quadrupole formula and considering the electromagnetic field in the near zone of the dipole, where most of the energy of the field is located. The results are discussed in Sec. 3 and compared with other sources of radiation, electromagnetic or gravitational, with a focus on stars such as magnetars 6 that possess extremely powerful magnetic fields.
2. Radiation from the near zone
Our starting point is the formula for the metric hijTT in the TT gauge (see Maggiore 7 for notation and details),
hijTT=1r2Gc4Λij, kln^Q¨klt-r∕c
(1)
where dots represent derivation with respect to time t,
Λij, kln^=PikPjl-12PijPkl ,
(2)
and
Pijn^=δij-n^in^j
(3)
is the projection tensor with respect to the unit vector n^. The quadrupole is defined as Qij=Mij-(1/3)δijMnn in terms of
Mij=1c2∫T00xixj dV,
(4)
Where T
00 is the 00 component of the energy-momentum tensor.
The energy radiated in the form of GWs in the direction of n^ is
dEdΩ=r2c332πG∫-∞∞dt h˙ijTTh˙ijTT.
(5)
2.1. Magnetic dipole
Consider the field of a magnetic dipole of magnitude m. In the near zone, the electric field can be neglected and the energy density of the magnetic field is
T00=18πm2r61+3r^∙u^t2 ,
(6)
where u^t is the unit vector in the direction of the dipole, and r^ is a unit radial vector. Further corrections to the electromagnetic field are of order ωr/c with respect to Bi (where ω is the rotation frequency of the dipole). For neutron stars of radius R ~ 10 km, the approximation is valid for ω≪c∕R∼3×104s-1.
It follows with some straightforward algebra that
Qij(t)=m25Rc2(u^i(t)u^j(t)-13δij),
(7)
where R is a lower cut-off that can be identified with the radius of the star. It is understood that the volume integral (4) covers the region r≥R.
The metric hijTT follows from the above formula and Eq. (1):
hijTT=1r2Gm25Rc6Λij,kld2dt2(u^k(tr)u^l(tr)),
(8)
where t
r
= t - r/c.
Let us now take a coordinate system in which the rotation axis of the dipole is in the z direction. Thus
u^t=u⊥ cosωt,u⊥ sinωt, u∥ ,
(9)
where u∥ is the constant component of u^t along the rotation axis and u⊥2=1-u∥2. In this same system of coordinates we can define the three orthonormal vectors
n^=sin θ cos ϕ, sin θ sin ϕ, cos θ
θ^=cos θ cos ϕ, cos θ sin ϕ, -sin θ
ϕ^=-sin ϕ, cosϕ, 0
(10)
together with the useful formulas
θ^iθ^jΛij,kl=θ^kθ^l-12Pkl,ϕ^iϕ^jΛij,kl=ϕ^kϕ^l-12Pkl,ϕ^iθ^jΛij,kl=ϕ^kθ^l.
(11)
2.2. Metric
The two metric potentials of the GW can be calculated from Eq. (8) and the formulas (11). The result is
h+≡hijTTϕ^iϕ^j=-hijTTθ^iθ^j=1rGm25Rc6d2dt2uϕ2-uθ2
h×≡-hijTTθ^iϕ^j=-1r2Gm25Rc6d2dt2uθuϕ
(12)
where uθ= u^∙θ^ and uϕ= u^∙ϕ^, and u^∙n^2+uθ2+uϕ2=1. The above two formulas can be written as
h++ih×=1rGm25Rc6d2dt2(uϕ-iuθ)2.
(13)
Explicitly
uϕ=u⊥ sinωt'
uθ=u⊥ cos θ cos ωt' - u∥ sin θ,
(14)
with ωt'=ωtr-ϕ, from where it follows that
h++ih×=1rGm25Rc6ω2u⊥2u⊥1+cos2θ cos2ωt'+2i cos θ sin 2ωt'-2u∥ sin θ × cosθ cosωt'+i sinωt' .
(15)
Accordingly, the spectrum of the GW has two lines, one at ω corresponding to the u∥ component, and one at 2ω corresponding to the u⊥ component (only the latter is present for a GW propagating along the rotation axis). The amplitude of the wave is of order Gm2ω2u⊥/(Rc6r).
2.3. Radiated energy
The radiated power can be calculated noticing that h˙ijTTh˙ijTT=|h˙++ih˙×|2. The energy radiated per unit time follows from Eq. (5) performing the integration over one period T=2π/ω and dividing by T. The result is
dPdΩ=Gm4200R2c9u⊥2ω6×1-cos4θ+u⊥25 cos4θ+24cos2θ+3 .
(16)
Finally, an integration over solid angles yields the total power radiated:
P=πGm475R2c9u⊥2ω6(1+18u⊥2).
(17)
3. Comparisons and conclusions
For the dipole field, we can set m=B0R3, where B
0 is the average strength of the magnetic field at the surface of the star. If B0∼1012G and ω∼1s-1, the power radiated in the form of gravitational radiation is
P~106B01012G4R10km10ω s6u⊥2 ergs∕s
according to formula (17). Of course, for average pulsars, this is many orders of magnitude below the power emitted in the form of electromagnetic waves, which is typically 1028 ergs/s 1. Nevertheless, for a millisecond magnetar with B0∼1014 G, the power of the GWs could be of the order of 1032 ergs/s.
It is also instructive to compare our results with those obtained by Bonazzola and Gourgoulhon 3 for the emission of GWs from the interior of a rotating neutron star. These authors obtained a value for the amplitudes of GWs
|h++ih×|∼1r4Gc4Iϵω2,
(18)
where I is the moment of inertia and ϵ is the ellipticity of the star; typical values of these parameters are ϵ∼10-6 or smaller, and I∼1045 g cm2. If we compare their result with our Eq. (15), we see that the amplitudes of the GWs produced by the rotating fluid are larger by a factor
1013ϵB0∕1012G-2
than the amplitudes of GWs produced by the rotation of the magnetic field. Thus, for a usual neutron star with B0∼1012 G, the contribution of the rotating dipole is comparatively negligible. However, it is not negligible for magnetars having fields B0∼1014 G 6 and rather small deformations ϵ<10-6.
In conclusion, the external magnetic field of a magnetar can make a significant correction to the gravitational radiation produced by the internal magnetized fluid.
References
1. F. Pacini, Nature 216 (1967) 567.
[ Links ]
2. S. Hacyan, Phys. Rev. D 93 (2016) 044066.
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3. S. Bonazzola and E. Gourgoulhon, Astron. Astrophys. 312 (1996) 675.
[ Links ]
4. C. Cutler, Phys. Rev. D 66 (2002) 084025.
[ Links ]
5. D.I. Jones, Class. Quan. Grav. 19 (2002) 1255.
[ Links ]
6. S. A. Olausen and V. M. Kaspi, Astrophys. J., Supp. Series 212 (2014) 1.
[ Links ]
7. M. Maggiore, Gravitational Waves (Oxford, Oxford U. Press, 2008) Vol. 1. Sec. 3.1.
[ Links ]
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